+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground/notation/relations/runion_3.ma".
-include "ground/xoa/or_3.ma".
-include "ground/xoa/ex_3_2.ma".
-include "ground/relocation/gr_tl.ma".
-
-(* RELATIONAL UNION FOR GENERIC RELOCATION MAPS *****************************)
-
-(*** sor *)
-coinductive gr_sor: relation3 gr_map gr_map gr_map ≝
-(*** sor_pp *)
-| gr_sor_push_bi (f1) (f2) (f) (g1) (g2) (g):
- gr_sor f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → gr_sor g1 g2 g
-(*** sor_np *)
-| gr_sor_next_push (f1) (f2) (f) (g1) (g2) (g):
- gr_sor f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → gr_sor g1 g2 g
-(*** sor_pn *)
-| gr_sor_push_next (f1) (f2) (f) (g1) (g2) (g):
- gr_sor f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → gr_sor g1 g2 g
-(*** sor_nn *)
-| gr_sor_next_bi (f1) (f2) (f) (g1) (g2) (g):
- gr_sor f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → gr_sor g1 g2 g
-.
-
-interpretation
- "relational union (generic relocation maps)"
- 'RUnion f1 f2 f = (gr_sor f1 f2 f).
-
-(* Basic constructions ******************************************************)
-
-(*** sor_idem *)
-corec lemma gr_sor_idem:
- ∀f. f ⋓ f ≘ f.
-#f cases (gr_map_split_tl f) #H
-[ @(gr_sor_push_bi … H H H)
-| @(gr_sor_next_bi … H H H)
-] -H //
-qed.
-
-(*** sor_comm *)
-corec lemma gr_sor_comm:
- ∀f1,f2,f. f1 ⋓ f2 ≘ f → f2 ⋓ f1 ≘ f.
-#f1 #f2 #f * -f1 -f2 -f
-#f1 #f2 #f #g1 #g2 #g #Hf * * * -g1 -g2 -g
-[ @gr_sor_push_bi | @gr_sor_push_next | @gr_sor_next_push | @gr_sor_next_bi ] /2 width=7 by/
-qed-.
-
-(* Basic inversions *********************************************************)
-
-(*** sor_inv_ppx *)
-lemma gr_sor_inv_push_bi:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
- ∃∃f. f1 ⋓ f2 ≘ f & ⫯f = g.
-#g1 #g2 #g * -g1 -g2 -g
-#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
-try (>(eq_inv_gr_push_bi … Hx1) -x1) try (>(eq_inv_gr_next_bi … Hx1) -x1)
-try elim (eq_inv_gr_push_next … Hx1) try elim (eq_inv_gr_next_push … Hx1)
-try (>(eq_inv_gr_push_bi … Hx2) -x2) try (>(eq_inv_gr_next_bi … Hx2) -x2)
-try elim (eq_inv_gr_push_next … Hx2) try elim (eq_inv_gr_next_push … Hx2)
-/2 width=3 by ex2_intro/
-qed-.
-
-(*** sor_inv_npx *)
-lemma gr_sor_inv_next_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
- ∃∃f. f1 ⋓ f2 ≘ f & ↑f = g.
-#g1 #g2 #g * -g1 -g2 -g
-#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
-try (>(eq_inv_gr_push_bi … Hx1) -x1) try (>(eq_inv_gr_next_bi … Hx1) -x1)
-try elim (eq_inv_gr_push_next … Hx1) try elim (eq_inv_gr_next_push … Hx1)
-try (>(eq_inv_gr_push_bi … Hx2) -x2) try (>(eq_inv_gr_next_bi … Hx2) -x2)
-try elim (eq_inv_gr_push_next … Hx2) try elim (eq_inv_gr_next_push … Hx2)
-/2 width=3 by ex2_intro/
-qed-.
-
-(*** sor_inv_pnx *)
-lemma gr_sor_inv_push_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
- ∃∃f. f1 ⋓ f2 ≘ f & ↑f = g.
-#g1 #g2 #g * -g1 -g2 -g
-#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
-try (>(eq_inv_gr_push_bi … Hx1) -x1) try (>(eq_inv_gr_next_bi … Hx1) -x1)
-try elim (eq_inv_gr_push_next … Hx1) try elim (eq_inv_gr_next_push … Hx1)
-try (>(eq_inv_gr_push_bi … Hx2) -x2) try (>(eq_inv_gr_next_bi … Hx2) -x2)
-try elim (eq_inv_gr_push_next … Hx2) try elim (eq_inv_gr_next_push … Hx2)
-/2 width=3 by ex2_intro/
-qed-.
-
-(*** sor_inv_nnx *)
-lemma gr_sor_inv_next_bi:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
- ∃∃f. f1 ⋓ f2 ≘ f & ↑f = g.
-#g1 #g2 #g * -g1 -g2 -g
-#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
-try (>(eq_inv_gr_push_bi … Hx1) -x1) try (>(eq_inv_gr_next_bi … Hx1) -x1)
-try elim (eq_inv_gr_push_next … Hx1) try elim (eq_inv_gr_next_push … Hx1)
-try (>(eq_inv_gr_push_bi … Hx2) -x2) try (>(eq_inv_gr_next_bi … Hx2) -x2)
-try elim (eq_inv_gr_push_next … Hx2) try elim (eq_inv_gr_next_push … Hx2)
-/2 width=3 by ex2_intro/
-qed-.
-
-(* Advanced inversions ******************************************************)
-
-(*** sor_inv_ppn *)
-lemma gr_sor_inv_push_bi_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
-#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
-elim (gr_sor_inv_push_bi … H … H1 H2) -g1 -g2 #x #_ #H destruct
-/2 width=3 by eq_inv_gr_push_next/
-qed-.
-
-(*** sor_inv_nxp *)
-lemma gr_sor_inv_next_sn_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f. ↑f1 = g1 → ⫯f = g → ⊥.
-#g1 #g2 #g #H #f1 #f #H1 #H0
-elim (gr_map_split_tl g2) #H2
-[ elim (gr_sor_inv_next_push … H … H1 H2)
-| elim (gr_sor_inv_next_bi … H … H1 H2)
-] -g1 #x #H
-/2 width=3 by eq_inv_gr_next_push/
-qed-.
-
-(*** sor_inv_xnp *)
-lemma gr_sor_inv_next_dx_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f2,f. ↑f2 = g2 → ⫯f = g → ⊥.
-#g1 #g2 #g #H #f2 #f #H2 #H0
-elim (gr_map_split_tl g1) #H1
-[ elim (gr_sor_inv_push_next … H … H1 H2)
-| elim (gr_sor_inv_next_bi … H … H1 H2)
-] -g2 #x #H
-/2 width=3 by eq_inv_gr_next_push/
-qed-.
-
-(*** sor_inv_ppp *)
-lemma gr_sor_inv_push_bi_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≘ f.
-#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
-elim (gr_sor_inv_push_bi … H … H1 H2) -g1 -g2 #x #Hx #H destruct
-<(eq_inv_gr_push_bi … H) -f //
-qed-.
-
-(*** sor_inv_npn *)
-lemma gr_sor_inv_next_push_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → f1 ⋓ f2 ≘ f.
-#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
-elim (gr_sor_inv_next_push … H … H1 H2) -g1 -g2 #x #Hx #H destruct
-<(eq_inv_gr_next_bi … H) -f //
-qed-.
-
-(*** sor_inv_pnn *)
-lemma gr_sor_inv_push_next_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≘ f.
-#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
-elim (gr_sor_inv_push_next … H … H1 H2) -g1 -g2 #x #Hx #H destruct
-<(eq_inv_gr_next_bi … H) -f //
-qed-.
-
-(*** sor_inv_nnn *)
-lemma gr_sor_inv_next_bi_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≘ f.
-#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
-elim (gr_sor_inv_next_bi … H … H1 H2) -g1 -g2 #x #Hx #H destruct
-<(eq_inv_gr_next_bi … H) -f //
-qed-.
-
-(*** sor_inv_pxp *)
-lemma gr_sor_inv_push_sn_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f. ⫯f1 = g1 → ⫯f = g →
- ∃∃f2. f1 ⋓ f2 ≘ f & ⫯f2 = g2.
-#g1 #g2 #g #H #f1 #f #H1 #H0
-elim (gr_map_split_tl g2) #H2
-[ /3 width=7 by gr_sor_inv_push_bi_push, ex2_intro/
-| elim (gr_sor_inv_next_dx_push … H … H2 H0)
-]
-qed-.
-
-(*** sor_inv_xpp *)
-lemma gr_sor_inv_push_dx_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f2,f. ⫯f2 = g2 → ⫯f = g →
- ∃∃f1. f1 ⋓ f2 ≘ f & ⫯f1 = g1.
-#g1 #g2 #g #H #f2 #f #H2 #H0
-elim (gr_map_split_tl g1) #H1
-[ /3 width=7 by gr_sor_inv_push_bi_push, ex2_intro/
-| elim (gr_sor_inv_next_sn_push … H … H1 H0)
-]
-qed-.
-
-(*** sor_inv_pxn *)
-lemma gr_sor_inv_push_sn_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f. ⫯f1 = g1 → ↑f = g →
- ∃∃f2. f1 ⋓ f2 ≘ f & ↑f2 = g2.
-#g1 #g2 #g #H #f1 #f #H1 #H0
-elim (gr_map_split_tl g2) #H2
-[ elim (gr_sor_inv_push_bi_next … H … H1 H2 H0)
-| /3 width=7 by gr_sor_inv_push_next_next, ex2_intro/
-]
-qed-.
-
-(*** sor_inv_xpn *)
-lemma gr_sor_inv_push_dx_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f2,f. ⫯f2 = g2 → ↑f = g →
- ∃∃f1. f1 ⋓ f2 ≘ f & ↑f1 = g1.
-#g1 #g2 #g #H #f2 #f #H2 #H0
-elim (gr_map_split_tl g1) #H1
-[ elim (gr_sor_inv_push_bi_next … H … H1 H2 H0)
-| /3 width=7 by gr_sor_inv_next_push_next, ex2_intro/
-]
-qed-.
-
-(*** sor_inv_xxp *)
-lemma gr_sor_inv_push:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f. ⫯f = g →
- ∃∃f1,f2. f1 ⋓ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2.
-#g1 #g2 #g #H #f #H0
-elim (gr_map_split_tl g1) #H1
-[ elim (gr_sor_inv_push_sn_push … H … H1 H0) -g /2 width=5 by ex3_2_intro/
-| elim (gr_sor_inv_next_sn_push … H … H1 H0)
-]
-qed-.
-
-(*** sor_inv_nxn *)
-lemma gr_sor_inv_next_sn_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f1,f. ↑f1 = g1 → ↑f = g →
- ∨∨ ∃∃f2. f1 ⋓ f2 ≘ f & ⫯f2 = g2
- | ∃∃f2. f1 ⋓ f2 ≘ f & ↑f2 = g2.
-#g1 #g2 elim (gr_map_split_tl g2)
-/4 width=7 by gr_sor_inv_next_push_next, gr_sor_inv_next_bi_next, ex2_intro, or_intror, or_introl/
-qed-.
-
-(*** sor_inv_xnn *)
-lemma gr_sor_inv_next_dx_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g →
- ∀f2,f. ↑f2 = g2 → ↑f = g →
- ∨∨ ∃∃f1. f1 ⋓ f2 ≘ f & ⫯f1 = g1
- | ∃∃f1. f1 ⋓ f2 ≘ f & ↑f1 = g1.
-#g1 elim (gr_map_split_tl g1)
-/4 width=7 by gr_sor_inv_push_next_next, gr_sor_inv_next_bi_next, ex2_intro, or_intror, or_introl/
-qed-.
-
-(*** sor_inv_xxn *)
-lemma gr_sor_inv_next:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f. ↑f = g →
- ∨∨ ∃∃f1,f2. f1 ⋓ f2 ≘ f & ↑f1 = g1 & ⫯f2 = g2
- | ∃∃f1,f2. f1 ⋓ f2 ≘ f & ⫯f1 = g1 & ↑f2 = g2
- | ∃∃f1,f2. f1 ⋓ f2 ≘ f & ↑f1 = g1 & ↑f2 = g2.
-#g1 #g2 #g #H #f #H0
-elim (gr_map_split_tl g1) #H1
-[ elim (gr_sor_inv_push_sn_next … H … H1 H0) -g
- /3 width=5 by or3_intro1, ex3_2_intro/
-| elim (gr_sor_inv_next_sn_next … H … H1 H0) -g *
- /3 width=5 by or3_intro0, or3_intro2, ex3_2_intro/
-]
-qed-.
-
-(* Constructions with gr_tl *************************************************)
-
-(*** sor_tl *)
-lemma gr_sor_tl:
- ∀f1,f2,f. f1 ⋓ f2 ≘ f → ⫰f1 ⋓ ⫰f2 ≘ ⫰f.
-#f1 cases (gr_map_split_tl f1) #H1
-#f2 cases (gr_map_split_tl f2) #H2
-#f #Hf
-[ cases (gr_sor_inv_push_bi … Hf … H1 H2)
-| cases (gr_sor_inv_push_next … Hf … H1 H2)
-| cases (gr_sor_inv_next_push … Hf … H1 H2)
-| cases (gr_sor_inv_next_bi … Hf … H1 H2)
-] -Hf #g #Hg #H destruct //
-qed.
-
-(*** sor_xxn_tl *)
-lemma gr_sor_next_tl:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f. ↑f = g →
- (∃∃f1,f2. f1 ⋓ f2 ≘ f & ↑f1 = g1 & ⫰g2 = f2) ∨
- (∃∃f1,f2. f1 ⋓ f2 ≘ f & ⫰g1 = f1 & ↑f2 = g2).
-#g1 #g2 #g #H #f #H0 elim (gr_sor_inv_next … H … H0) -H -H0 *
-/3 width=5 by ex3_2_intro, or_introl, or_intror/
-qed-.
-
-(*** sor_xnx_tl *)
-lemma gr_sor_next_dx_tl:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f2. ↑f2 = g2 →
- ∃∃f1,f. f1 ⋓ f2 ≘ f & ⫰g1 = f1 & ↑f = g.
-#g1 elim (gr_map_split_tl g1) #H1 #g2 #g #H #f2 #H2
-[ elim (gr_sor_inv_push_next … H … H1 H2)
-| elim (gr_sor_inv_next_bi … H … H1 H2)
-] -g2
-/3 width=5 by ex3_2_intro/
-qed-.
-
-(*** sor_nxx_tl *)
-lemma gr_sor_next_sn_tl:
- ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1. ↑f1 = g1 →
- ∃∃f2,f. f1 ⋓ f2 ≘ f & ⫰g2 = f2 & ↑f = g.
-#g1 #g2 elim (gr_map_split_tl g2) #H2 #g #H #f1 #H1
-[ elim (gr_sor_inv_next_push … H … H1 H2)
-| elim (gr_sor_inv_next_bi … H … H1 H2)
-] -g1
-/3 width=5 by ex3_2_intro/
-qed-.
-
-(* Inversions with gr_tl ****************************************************)
-
-(*** sor_inv_tl_sn *)
-lemma gr_sor_inv_tl_sn:
- ∀f1,f2,f. ⫰f1 ⋓ f2 ≘ f → f1 ⋓ ↑f2 ≘ ↑f.
-#f1 #f2 #f elim (gr_map_split_tl f1)
-/2 width=7 by gr_sor_push_next, gr_sor_next_bi/
-qed-.
-
-(*** sor_inv_tl_dx *)
-lemma gr_sor_inv_tl_dx:
- ∀f1,f2,f. f1 ⋓ ⫰f2 ≘ f → ↑f1 ⋓ f2 ≘ ↑f.
-#f1 #f2 #f elim (gr_map_split_tl f2)
-/2 width=7 by gr_sor_next_push, gr_sor_next_bi/
-qed-.
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